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Cite this: SoftMatter, 2016,

12, 3963

Depletion controlled surface deposition ofuncharged colloidal spheres from stablebulk dispersions

Samia Ouhajji,*ab Tommy Nylander,b Lennart Piculell,b Remco Tuinier,ac Per Linseb

and Albert P. Philipse*a

The competition between surface adsorption and bulk aggregation was investigated for silica colloids

dispersed in cyclohexane in contact with hydrophobized silica substrates. Central to this study is that the

colloids and surfaces have the same material and surface properties. Colloid–colloid and colloid–surface

interactions were controlled by addition of polymers providing depletion interaction. Bulk instability was

determined by turbidity and viscosity measurements and surface adsorption by ellipsometry measurements.

At increasing polymer concentration, strong surface adsorption occurred at polymer concentrations below

that required for bulk phase separation. Complementary Monte Carlo simulations with the use of a new

weak depletion theory support quantitatively the experimental observation of the existence of an interval of

interaction strength at which aggregation in bulk is negligible while surface adsorption is substantial.

1 Introduction

Surface coatings can be made by particle adsorption fromsolutions or dispersions onto macroscopic substrates.1,2 Anexample of the latter is the convective assembly used for thefabrication of colloidal crystals.3–5 Adsorption from solutions ordispersions has the advantage that adsorption rates can betuned by varying parameters such as concentration andtemperature.

Generally, the intention is to obtain homogeneous surfacecoatings. However, to achieve an adsorbed layer with close-packed particles is far from trivial. For instance, if particle–particle attractions are too strong, particles may aggregatebefore adsorbing onto a surface, contributing to inhomo-geneous coatings. Particle deposition on surfaces is thus con-trolled by the delicate balance between particle–particle andparticle–surface interactions.

Extensive literature can be found on the competitionbetween bulk aggregation and surface adsorption,6–12 appearingwhen a particle dispersion is in contact with a surface. Wijtinget al.13 investigated the wetting behaviour in a system of

organophilic silica spheres in cyclohexane with poly(dimethyl-siloxane) (PDMS) as depletant. At sufficiently high concentra-tions of both colloids and polymers, a (colloidal) gas–liquidphase separated system was obtained. Upon approach of thecritical point from the two-phase region, the contact angle of thegas–liquid interface on various substrates changed from partialwetting to complete wetting.

Furthermore, fluid–solid transitions on solid substrateswere studied with optical microscopy, and a fluid monolayerof large charge-stabilized polystyrene spheres in water at a hardwall was observed at concentrations below the bulk phaseboundary.14,15 As the concentration of depletant spheresincreased, the fluid surface layer became solid, still in coexistencewith the bulk fluid. In binary mixtures of colloidal spheres, asurface phase that phase separated before the bulk, purely drivenby entropy, was observed.16 In a similar system, the occurrence ofa wall crystal–fluid coexistence was reported.17

In a recent study18 using a thermodynamic chemical equilibriummodel and Monte Carlo simulations, Linse and Wennerstrompredict the existence of an interval of interaction strength wheresurface adsorption is significant while bulk instability throughnucleation remains negligible. Two assumptions are central in thisstudy,18 viz. (i) the particles and substrates are made of the samematerial, and (ii) the interaction range is much smaller than theparticle size allowing the Derjaguin approximation to be applied.Under these assumptions the particle–surface interaction becomestwice as strong as the particle–particle interaction. At increasingattraction, the thermodynamic model predicted the followingscenarios to occur: (A) weak particle adsorption onto the surface,

a Van’t Hoff Laboratory for Physical and Colloid Chemistry,

Debye Institute for Nanomaterials Science, Utrecht University, Padualaan 8,

3584 CH, The Netherlands. E-mail: S.Ouhajji@uu.nl, A.P.Philipse@uu.nlb Physical Chemistry, Department of Chemistry, Lund University, P.O. Box 124,

S-22100 Lund, Swedenc Laboratory of Physical Chemistry, Department of Chemical Engineering and

Chemistry and Institute for Complex Molecular Systems, Eindhoven University of

Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Received 27th November 2015,Accepted 10th March 2016

DOI: 10.1039/c5sm02892b

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(B) particle association on the surface forming a denser singleadsorbed layer, (C) formation of a second adsorbed layer on thesurface, (D) multiple adsorbed layers on the surface and (E) bulkphase separation.

None of the experimental studies mentioned above, how-ever, has reported results that allow quantitative comparison tothe Linse–Wennerstrom scenario18 and its key prediction ofsignificant surface adsorption from a stable hard-sphere fluid.The aim of this paper is to quantitatively test this predictionusing the hard-sphere silica colloids developed by Vrij andco-workers.19 The effective particle–particle and particle–surface attractions were tuned by adding PDMS as depletantat varying concentrations. The occurrence of bulk aggregationwas determined both with turbidity and dynamic light scatter-ing measurements, and surface adsorption from particle dis-persion was quantified by ellipsometry. Simulations incombination with recently developed weak-depletion theory20

were performed to further underpin and explain our experi-mental results.

2 Experimental2.1 Materials

2.1.1 Chemicals. Cyclohexane (99%, Sigma-Aldrich) andethanol (99.7%, Solveco) were used as solvent. Silica substrateswere rinsed with hydrogen peroxide (30% by weight H2O,Honeywell Burdick and Jackson), ammonia solution (E25%NH3, Honeywell Burdick and Jackson) and hydrochloric acid(fuming, 37%, Merck). The substrates were hydrophobizedusing 1-octadecanol (99%, Honeywell Burdick and Jackson)dissolved in chloroform (for HPLC, Z99.9%, Sigma-Aldrich).Poly(dimethylsiloxane) (PDMS, Fluka) with a weight averagemolar mass of 95 000 g mol�1 and a polydispersity index of 1.9was used as depletion polymer. The cuvette for ellipsometry wascleaned with 5 v/v% Decon 90 (Decon laboratories) in water.

The cuvette for dynamic light scattering was cleaned withHellmanex (Hellma). All chemicals were used as received andstored at ambient conditions except for hydrogen peroxidewhich was stored in a refrigerator. All water used in theexperiments was purified by a Milli-Q system.

2.1.2 Colloidal system. Silica spheres stabilized with1-octadecanol in cyclohexane were prepared as describedearlier.19,21 The silica dispersion had been prepared twodecades ago22 and clearly has demonstrated a long-termcolloidal stability. The van der Waals attraction between twooctadecyl-coated silica spheres is minimized because the refrac-tive indices of silica and cyclohexane are closely matched,23 andhence the interaction between two colloidal spheres is wellrepresented by a hard-sphere potential. In addition, cyclo-hexane is a good solvent for the octadecyl coating of thespheres. Transmission electron microscopy (TEM) images(Fig. 1) were used to determine the shape of the particles(Philips TECNAI 12). Due to their small size, the dispersedsilica particles are not completely spherical (see Fig. 1). Theapparent hydrodynamic diameter of these spheres was 74 �14 nm, as determined by dynamic light scattering. PDMS24,25

with a radius of gyration Rg = 14.4 nm was used as depletant.The radius of gyration of PDMS was calculated followingVincent26,27 (see Appendix A). The y-temperature of PDMS incyclohexane is �80 1C, so the polymer is in a good solvent atroom temperature.28 No indications were found that PDMSsignificantly adsorbs onto the silica particles, see Fig. 6. How-ever, as discussed in Section 4, PDMS is not fully repelled by thesilica surfaces.

The phase diagram of the studied system was obtained bymixing varying concentrations of the silica spheres and PDMS.Phase separation, determined by visual inspection of thesamples, occurred at sufficiently high concentrations of bothcolloidal particles and polymer and was typically observedwithin 30 minutes after mixing the two species. The polymer-to-colloid size ratio Rg/R, with R the hydrodynamic radius ofthe colloids, is 0.38, which in theory allows for colloidal gas,liquid and crystalline phases.29 However, only gas–liquid phaseseparation was observed in the present study.

2.1.3 Substrates. Hydrophobized silica substrates were pre-pared from 4 inch polished silicon wafers (p-type, boron doped,resistivity of 1–10 O cm) with a thermally oxidized silica layer ofaround 300 Å, purchased from Semiconductor Wafer Inc.,Taiwan. The oxidized silicon wafer was cut into small slideswith a width of about 1 cm and a length of 2–3 cm. Prior to use,the substrates were cleaned in an alkaline mixture of NH4OH,H2O2 and H2O (3 : 3 : 16 by volume) at 80 1C for 5 minutes andrinsed with water, followed by an acid mixture consisting ofHCl, H2O2 and H2O (3 : 3 : 16 by volume) at 80 1C for 10 minutes,after which they were thoroughly rinsed with water and thenwith ethanol. Finally, the substrates were stored in ethanol.

The substrates were hydrophobized in a melt of 1-octadecanol.Prior to hydrophobization, the silica substrates were rinsedwith ethanol, dried with nitrogen and plasma cleaned in lowpressure air at a pressure of 0.0001 mbar for 5 minutes (HarrickScientific Corp Plasma Cleaner, Model PDC-3XG) to remove any

Fig. 1 Typical TEM image of C18-coated silica particles with an averagediameter of 64� 5 nm and apparent hydrodynamic diameter of 74� 14 nm.

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organic impurities. In a two-neck round-bottom flask, 30 gramsof 1-octadecanol was heated to 180–200 1C while stirring with amechanical stirrer. A silica substrate was placed in this melt for3–4 hours. The hydrophobized substrate was rinsed three timesin pre-heated chloroform at about 55 1C, sonicated in ethanolfor 15 minutes and, finally, stored in ethanol. Before ellipso-metry experiments, substrates were rinsed with ethanol anddried with nitrogen.

2.2 Methods

2.2.1 Ellipsometry. Surface adsorption was monitoredin situ by null ellipsometry measurements. A modified, auto-mated Rudolph research thin-film null ellipsometer (model43603-200E) was used, as described in detail by Tiberg andLandgren.30 The instrument was equipped with high-precisionstep motors and controlled by a personal computer. A xenon arclamp, filtered to a wavelength of 401.5 nm, was used as the lightsource. Measurements were performed at an angle of incidenceof 67.871. The substrate was mounted vertically in a 7 mLtrapezoid cuvette made of optical glass and thermostated to25 1C. A magnetic stirrer agitated the sample at about 300 rpm.An inlet and an outlet tube in the cuvette were connected to aHPLC pump and a multichannel peristaltic pump, respectively,in order to change the solution without emptying the cuvette.Due to the volatile nature of the solvent, additional cyclohexanewas added every minute by the HPLC pump to maintain aconstant sample volume.

Prior to each experiment, the refractive index and the thick-ness of the silica substrate were determined using a three-layermodel of bulk silicon with a silica layer in a surroundingmedium. To this end, the ellipsometric angles C and D weremeasured in two different media and corrections were made foroptical imperfections of the instrument by averaging over fourzones. Typical values of the refractive indices were 5.5 � 0.3j forbulk silicon and 1.48 for the thermally grown silicon oxide filmwith a thickness of 300 � 20 Å.

After characterization of the optical properties of the sub-strate with the oxide layer, the thickness and refractive index ofan additional adsorbed layer are obtained using a four-layermodel from recorded changes in C and D as a function of time.From these values, the adsorbed amount G can be calculated asa function of the bulk concentration, using de Feijter’s approxi-mation,31 according to:

G ¼ dfnf � n0

dn=dc(1)

with df being the thickness and nf the refractive index of theadsorbed film, n0 the refractive index of the solvent and dn/dcthe refractive index increment of the adsorbed substance. Therefractive index increment of the silica dispersion was measuredusing a multi-wavelength refractometer (Abbe 60/ED fromBellingham and Stanley Ltd) and had a value of 0.00973 mL g�1.

Concentrated silica dispersion was added to about 5 mLcyclohexane in the cuvette to a final silica concentration of3 volume percent followed by addition of PDMS with varyingconcentrations (0–14 mg mL�1).

2.2.2 Dynamic light scattering. The apparent hydro-dynamic diameter of the silica spheres in absence and presenceof depletant was measured by dynamic light scattering using aMalvern Instruments Zetasizer Nano-ZS. A quartz cuvette wasfilled with 2 mL samples containing 3 v/v% silica particles and0–14 mg mL�1 PDMS in cyclohexane. Prior to the measure-ments, the samples were filtered through a syringe filter (PTFEmembrane, 0.20 mm, Minisart). Measurements were performedat 25 1C after ten minutes of equilibration at an angle of incidenceof 1731. The obtained apparent hydrodynamic diameter wasaveraged over ten measurements.

2.2.3 Turbidity. Turbidities were determined by measuringthe transmittance T of 2 mL samples containing 3 v/v% silicaspheres and 0–14 mg mL�1 PDMS in cyclohexane using aVarian Cary WinUV spectrophotometer. The transmittance ofpure cyclohexane was measured as well. Measurements wereperformed in a thermostated quartz cuvette at 25 1C in thewavelength range of 200 to 800 nm every 0.500 nm at a scan rateof 300 nm min�1 with an average time of 0.1 s and a spectralbandwidth of 2.0 nm.

The relative turbidity was related to the transmittanceaccording to eqn (2):

Relative turbidity ¼ Tsolvent � Tsample

Tsolvent(2)

at wavelength 401.5 nm, the same as for the ellipsometerymeasurements.

3 Experimental results and discussion3.1 Colloids and substrates

The phase diagram of the colloid–polymer mixture is shown inFig. 2. Experiments were performed with samples containing3 v/v% silica spheres and 0–14 mg mL�1 PDMS, which wasfound to be in the one-phase region of the system.

The silica substrates were hydrophobized by the same methodas the silica spheres, namely in a melt of 1-octadecanol.

Fig. 2 Experimental and theoretical phase diagrams of mixtures of hydro-phobized silica spheres and PDMS. Above the phase boundary liquid–gasphase separation is observed. The compositions of the samples used in ourstudy are indicated by the vertical grey bar.

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Although the same treatment does not guarantee equal surfacedensity of octadecyl chains, it does ensure that the chemicalnature of the surfaces is the same. The water contact angle onthe silica substrates was 91.51 and gives an indication ofeffective hydrophobization. From AFM images (not shown) itwas deduced that the surfaces were overall smooth with roughnesson the order of a few nanometres. However, occasionally largesurface irregularities up to 200 nm could be found. These objectsare likely to be impurities like dust as the hydrophobic surfacetends to attract impurities. The substrates were therefore stored inethanol to reduce the risk of surface contamination.

We also prepared C18 silica substrates in a solution of1-octadecanol in triethyl phosphate.32 However, it was foundthat the surface layers were unstable, giving poor reproducibilityof the adsorption data. Possibly these effects stem from partialdissolution of the stearyl layer, resulting in less hydrophobic andless smooth surfaces.

3.2 Surface adsorption

A typical example of the adsorbed amount and normalizedthickness versus time from an ellipsometry experiment isshown in Fig. 3. In Fig. 4 and 5, the adsorbed amount andthe normalized thickness, respectively, are plotted as functionsof polymer concentration to illustrate the influence of thedepletant and thus the increase in attraction between theparticles. The thickness and the adsorbed amount are givenat three different times: 53 minutes and 88 minutes after theaddition of PDMS and 30 minutes after rinsing (bulk solutionreplaced by pure solvent). These values represent steady-stateconditions.

Fig. 4 shows that adsorption of particles onto a surfacearises already in the absence of depletant. This finite adsorp-tion is very likely a consequence of the van der Waals attractionbetween particles and the surface (see Appendix B). Uponaddition of PDMS up to and including a concentration of6 mg mL�1, the adsorbed amount shows a relatively weakincrease. Above a depletant concentration of 6 mg mL�1,

a steeper increase occurs that levels off again at 12 mg mL�1.No decrease in adsorbed amount is observed after rinsing,provided that the surface layers are produced at PDMS con-centrations of 6 mg mL�1 or less. However, a significantfraction is removed by rinsing with pure solvent from layersproduced at higher PDMS concentrations.

Fig. 5 shows that the normalized thickness follows a similar trendas the adsorbed amount, however, the relative variation with PDMSconcentration is much smaller than for the thickness. The thicknessincreases rapidly up to and including 4 mg mL�1, after which itattains basically a constant value between 4 and 8 mg mL�1 andfinally increases again above 8 mg mL�1 PDMS. The fact that thethickness remains constant while the adsorbed amount increasesindicates that a more densely packed layer is being formed.

The values of both the adsorbed amount and the normalizedthickness recorded after rinsing, increase slightly with theinitial PDMS concentration, but the variations are much lessthan before rinsing, see Fig. 4 and 5.

Fig. 3 Adsorbed amount and normalized thickness versus time for theadsorption of silica particles on a hydrophobized silica surface from a 3 v/v%silica dispersion with 10 mg mL�1 PDMS in cyclohexane as measured byellipsometry. The silica dispersion was added at zero time and rinsing wasperformed after 90 minutes (indicated by the green arrow).

Fig. 4 Adsorbed amount versus PDMS concentration for the adsorptionof silica particles on a hydrophobized silica surface from a 3 v/v% silicadispersion with added PDMS in cyclohexane after 53 minutes and88 minutes of adsorption and after 30 minutes of rinsing.

Fig. 5 Thickness of adsorbed layers versus PDMS concentration for theadsorption of silica particles on a hydrophobized silica surface from a 3 v/v%silica dispersion with added PDMS in cyclohexane after 53 minutes and88 minutes of adsorption and after 30 minutes of rinsing.

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We note in passing that in the theoretical study of depletiondriven sphere crystallization on a wall,17 the authors ‘expect asharp transition of the whole layer from fluid to crystal with ajump in density’. The much more gradual transition in ourexperiments possibly stems from the size polydispersity of oursilica spheres that suppresses crystallization. In future workthis issue could further be studied via computer simulations ofdepletion driven deposition for monodisperse and slightlypolydisperse spheres.

3.3 Bulk aggregation

Bulk aggregation was studied for the same samples as used inthe surface adsorption experiments. To this end, turbidity andapparent hydrodynamic diameter measurements were per-formed in the one-phase region of the phase diagram.

3.3.1 Turbidity. The turbidity of a 3 v/v% silica particledispersion versus concentration of PDMS in cyclohexane isshown in Fig. 6. The value of the relative turbidity remainsessentially constant over the entire range, fluctuating around0.5. This clearly shows that the particles do not aggregate in thebulk. Only at the polymer concentration of 14 mg mL�1, closeto the bulk phase boundary, a slight increase of the turbiditycan be observed.

3.3.2 Apparent hydrodynamic diameter. The apparenthydrodynamic diameter as a function of PDMS concentrationis shown in Fig. 6. The data are corrected for the increase of theviscosity of the medium due to the polymer. These resultsfurther demonstrate that no bulk aggregation takes place asthe hydrodynamic particle diameter hardly changes from about

70 nm over the range of polymer concentrations. A slightincrease in size can be observed at the highest depletantconcentration, see Fig. 6. In addition, these results confirmthat PDMS does not significantly adsorb onto the silica spheres.

More importantly, we have experimentally found the exis-tence of an interval of interaction strength, in which bulkaggregation is negligible and surface adsorption substantial.In the next section our experimental findings are directlycompared to theory and simulations using our experimentalparameters.

4 Comparison to theory andsimulations

The depletion interaction potential, Wdep(h), between twospheres and a sphere and a planar surface can be calculatedusing the adsorption method.29 The depletion layer is calculatedfrom the negative adsorption of polymer around a colloidalparticle33 instead of conventionally taking the depletion layerthickness to be equal to the polymer’s radius of gyration. Thisgives the depletion interaction potential:

WdepðhÞ ¼ �ðn0

1

n0@P@n0

� �½GðhÞ � Gð1Þ�dn0 (3)

where n denotes the polymer bulk concentration, P the osmoticpressure of the solution and G(h) the adsorbed amount ofpolymer segments at particle separation h. Moreover, the numberdensity n equals 3fp=ð4pRg

3Þ with Rg being the radius of

gyration and fp the relative polymer concentration, the latterdefined as unity at the overlap concentration. The osmoticcompressibility can be written as a function of fp as follows:33

@bP@n

� �¼ 1þ 2:63fp

1þ 3:25fp þ 4:15fp2

1þ 1:48fp

!0:309

(4)

where b = 1/kT. For two colloidal hard spheres with radius R in apolymer solution, the adsorbed amount is given by:33

GðhÞ � Gð1Þ

¼2

3pnDs

3 1� h

2Ds

� �2

2þ 3R

Dsþ h

2Ds

� �for h � 2Ds

0 for h4 2Ds

8><>:

(5)

with Ds being the depletion layer thickness around a sphere.For the interaction between a colloidal sphere and a flat wall,the adsorption follows from:33

Fig. 6 Turbidity (left y-axis) and apparent hydrodynamic diameter (righty-axis) versus PDMS concentration of a 3 v/v% silica dispersion with addedPDMS in cyclohexane. Corrections are made for the increase in sampleviscosity due to increasing polymer concentration.

GðhÞ � Gð1Þ ¼

4

3pn Rþ Dsð Þ3 for hoDw � 2R� Ds

1

3pn Ds þ Dw � hð Þ2 3Rþ 2Ds � Dw þ hð Þ for Dw � 2R� Ds � h � Dw þ Ds

0 for h4Dw þ Ds

8>>>>>><>>>>>>:

(6)

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with Dw being the depletion layer thickness near a single flatwall. The quantities Dw and Ds are given by respectively:29

Dw ¼1:07Rgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 3:95fp1:54

q (7)

Ds ¼ 0:865R

Rg

� �2

þ 3:95R

Rg

� �2

fp1:54

!�0:44R (8)

These classic or full sphere–sphere and sphere–wall depletionpotentials are shown in Fig. 7. The Derjaguin approximationholds reasonably well as the sphere–wall interaction is almosttwice as large as the sphere–sphere interaction.

However, we discovered that the depletion potential as out-lined above does not accurately describe the phase behaviour ofour system, see Fig. 2. Recently, the effect of weak depletion, i.e.when the probability of polymers residing at the surface of acolloidal particle is small but non-zero, was studied in detail byTuinier and co-workers.20 The weak depletion potential differsfrom the full depletion potential in the value for the depletionlayer thickness. As a result of the non-zero probability of thepolymer to be on the particle’s surface, the depletion layerthickness decreases with a factor of 0.71 in our system. Thischange has a large effect on the depletion interaction potentialas shown in Fig. 7. The effective range of the weak depletionpotential as well as the interaction energy at contact (h = 0) aresmaller for weak depletion. The phase diagram of our experi-mental system, shown in Fig. 2, can be predicted reasonably wellwhen taking weak depletion into account. Simulations of thephase behaviour and surface adsorption of our system wereperformed using both the full and the weak depletion potentials.

4.1 Simulation method

The onset of bulk aggregation was investigated using Metropo-lis Monte Carlo simulations using a fixed number of particlesN, volume V and temperature T (canonical ensemble).34 Theonset of surface adsorption in particle dispersions was examinedusing fixed chemical potential m, volume V and temperature T

(grand canonical ensemble). The canonical simulations wereperformed using a cubic box with edges (L) of 1920 nm andperiodic boundary conditions in all three dimensions, whereasthe grand canonical simulations were made using a box withedges of 1524.5 nm, 1524.5 nm, 3047 nm (XYZ) and periodicboundary conditions applied in the X and Y directions. Theparticles had a radius of 37 nm, the temperature 298 K wasused throughout, and N = 1000 particles at volume fraction of3% were used in the canonical simulations. The range of theinteraction potential was fixed to 1.38s with s being equal tothe particle diameter, corresponding to a polymer-to-colloidsize ratio of 0.38. The chemical potential of the particles wasdetermined from the bulk simulations using the Widom particleinsertion method.

The full and weak depletion interaction potentials as specifiedpreviously were used at selected values of relative polymerconcentration as given in Table 1. After equilibration the simula-tions involved at least 106 single particle trial moves per particlewith displacement parameter equal to 200 nm.

4.2 Simulation results

Fig. 8 shows the structure factor, S(q), obtained from simula-tions of bulk systems with a weak depletion potential atindicated values of fp. In the thermodynamic limit, the structurefactor of a phase-separated system diverges as q approaches 0.For a finite system of N particles, the structure factor of anunstable bulk system limits to N as q approaches 0. In Fig. 8, the

Fig. 7 Full and weak depletion potentials for sphere–sphere (ss) andsphere–wall (sw) interactions at a relative polymer concentration fp = 0.5.

Table 1 Overview of examined depletion potentials, relative polymerconcentrations fp and the obtained intervals at which the adsorptionthreshold, fp,surface*, and bulk instability, fp,bulk*, appear

Potential fp fp,surface* fp,bulk*

Full depletion 0.48 (0.02) 0.56 0.50–0.52 0.54–0.56Weak depletion 0.00, 1.40 (0.05) 1.60 1.40–1.45 1.55–1.60

Fig. 8 Structure factor S(q) versus the wave vector q at 3 v/v% particlesand at indicated polymer volume fractions, fp, from canonical simulationsof the bulk dispersions. S(q) diverges for a phase-separated system. Snap-shots from the end of simulations with a weak depletion potential areincluded for fp = 1.55 and fp = 1.60, as indicated by the arrows.

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lower limit of q is equal to 2p/L as a consequence of the imposedperiodic boundary conditions. The maximum q is slightly smallerthan a particle diameter. The structure factor at fp = 0 is relativelyfeatureless, characteristic for hard-sphere repulsion at low volumefraction. At fp = 1.60, higher-order maxima (of high-densityaggregates) appear and the structure factor diverges. Thus, bulkinstability starts to appear between fp = 1.55 and fp = 1.60. Thevalue of S(q = 2p/L) = 117 indicates that a large aggregate isformed. Snapshots of final configurations from the canonicalsimulations at fp = 1.55 and fp = 1.60 can also be found inFig. 8. It is clear that the bulk is stable at fp = 1.55 but aggregatesappear at (and above) fp = 1.60.

The particle number density distributions as a function of z,perpendicular to the surface from the grand canonical simula-tions with the weak depletion potential are shown in Fig. 9. Atthe relative polymer volume fraction of 1.40, an adsorbed layerof finite width is obtained and at most two particle layers can bedistinguished. However, at fp = 1.45 the depicted three sequen-tial runs indicate that the result has not yet converged. Theadsorbed layer remains growing. Snapshots of final configura-tions from the simulations of the surface systems for fp = 1.40and fp = 1.45(3) are included in Fig. 9. At fp = 1.45 theadsorbed layer contains more particles and is more denselypacked than at lower polymer volume fraction. It is expectedthat a homogeneous high density phase bridging the surfaceswould have been obtained at equilibrium. Thus, the surface-induced instability appears between fp = 1.40 and fp = 1.45,before the bulk instability.

The intervals of polymer volume fraction at which theonset of surface- and bulk-induced instability appear are sum-marized in Table 1. We conclude that the surface-inducedinstability appears at lower polymer volume fraction than thebulk instability.

For completeness, simulations were also performed withthe full depletion potential. The obtained intervals of surface-induced and bulk instabilities can be found in Table 1. Theinterval of interaction strength where surface adsorption issubstantial while the bulk remains one phase is almost non-existent. This contradicts our experimental results that clearlyshow that such an interval does indeed exist. Furthermore, ourresults confirm that weak depletion is essential for an accuraterepresentation of our experimental system and, consequently,that classical (full) depletion theory falls short in this case.Finally, the obtained experimental results serve as a validationof weak depletion theory20 for systems consisting of stearylsilica spheres in cyclohexane with PDMS as depletant.

5 Conclusions

We have shown that surface adsorption precedes bulk aggrega-tion for particles and substrates that have the same materialand surface properties. The strength of the particle–particleand particle–surface interactions could be controlled simulta-neously by addition of depletion polymer. Multiple layers ofparticles were found to adsorb onto the substrates as thedepletion interaction was increased. Monte Carlo simulationswith recently developed weak depletion theory accurately describethe phase behaviour of our system and quantitatively support theexperimental observations.

Our experimental and simulation results correspond wellwith predictions from theory and simulations by Linse andWennerstrom, and show that indeed an interval in the interactionstrength is present, at which surface adsorption is significantwhile bulk aggregation is negligible.

AppendicesAppendix A: radius of gyration of PDMS

The radius of gyration of PDMS was calculated followingVincent.26,27 The radius of gyration, Rg, of a depletion polymeris given by:

Rg = rogr0.1 (9)

Here, rog is the unperturbed value of the radius of gyration and r

the number of effective segments per polymer chain. r is linkedto the statistical segment length l through the extended polymerlength L:

L = rl (10)

r itself is defined by constants that are related to the nature ofthe polymer:

r ¼ 0:408s

AMs

� �2

Mw (11)

Here, Ms is the molar mass of the polymer repeat unit, A anempirical constant characteristic of the polymer chains in ay-solvent, s the length of each segment and Mw the molecularweight of the polymer. The unperturbed value of the radius of

Fig. 9 Number density distributions of particles near one surface, r(z),versus distance z from the surface at 3 v/v% particles and at indicatedpolymer volume fractions, fp, from grand canonical adsorption simula-tions. The surface is located at the right end of the graph. Equilibratedresult is shown for fp = 1.40. Three successive simulations (each 106

Monte Carlo trial moves) at fp = 1.45 show that this result has not yetconverged. Snapshots of simulation box containing two planar surfacesfrom the end of simulations with a weak depletion potential are includedfor fp = 1.40 and fp = 1.45(3), as indicated by the arrows.

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3970 | Soft Matter, 2016, 12, 3963--3971 This journal is©The Royal Society of Chemistry 2016

gyration, rog, can be calculated from:

rog ¼A2Ms

0:408s

ffiffirp

(12)

The values of A, Ms and s can be found in literature and have thefollowing values for PDMS:26

A ¼ 0:027 nmffiffiffiffiffiffiffiffimolp ffiffiffi

gp �1

Ms ¼ 74 g mol�1

s ¼ 0:25 nm

(13)

PDMS used here had a molecular weight of 95 000 g mol�1.Together, these values yield:

r = 248

rog = 8.32 nm (14)

The calculated value of the radius of gyration of PDMS with amolecular weight of 95 000 g mol�1 is:

Rg = rogr0.1 = 14.4 nm (15)

Appendix B: van der Waals attraction

The results in Section 3.2, show that the surface adsorption ofparticles increases with increasing depletant concentration andthus with increasing attraction between the particles. However,it was also found that surface adsorption occurs even in theabsence of depletion polymer. Therefore, the reference condi-tion of a pure hard-wall interaction18 is not strictly achieved inour model system in the absence of depletant. Adsorption inthe absence of depletion polymer very likely occurs as a con-sequence of the van der Waals attraction.

The van der Waals attraction Wss(h) between two identicalspheres with radius a at a small separation h between thesurfaces of the particles, is given by:35

WssðhÞ ¼ �A

12

a

h

h ifor h� a (16)

Here, A is the Hamaker constant. The sphere–wall attraction,Wsw(h) is twice as large as in eqn (16):

WswðhÞ ¼ �A

6

a

h

h ifor h� a (17)

Eqn (16) and (17) can be used to estimate van der Waalsattraction in stearyl silica–cyclohexane systems. The Hamakerconstant of stearyl silica in cyclohexane is about 0.15kT at atemperature of 298 K.36 Eqn (16) and (17) then become:

bWssðhÞ � �0:15

12

a

h

h i(18)

bWswðhÞ � �0:15

6

a

h

h i(19)

For a sphere radius a = 37 nm, bWss(h) and bWsw(h) arepresented graphically as a function of separation h in Fig. 10.A stearyl chain extends 1.5 nm in cyclohexane from the surface

of the silica particle.36 The minimum separation between twostearyl silica spheres (or one stearyl silica sphere and a stearylsilica wall) is thus about 3 nm. For our dispersion, the sphere–sphere van der Waals attraction has a value of 0.15kT ath = 3 nm. The sphere–wall van der Waals attraction is twiceas strong and has a value of 0.3kT.

This 0.3kT is a lower-bound as various factors can – and verylikely will – strengthen the sphere–wall attraction. Examples ofthese factors are non-spherical shape and surface irregularitiesthat increase the contact areas, and imperfections in the stearylcoating on the surfaces.

Acknowledgements

Prof. H. Wennerstrom is thanked for helpful discussions; SOalso thanks Dr T. Halthur and Dr M. Yanez Arteta for experi-mental support and acknowledges financial support from theErasmus programme for a stay at Lund University. LP, PL andTN acknowledge Swedish Foundation for Strategic Research(SSF) for financial support via framework grant RMA08-0056.

References

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Fig. 10 Sphere–sphere and sphere–wall van der Waals attraction versussurface-to-surface distance h of stearyl silica spheres dispersed in cyclo-hexane with a diameter of 74 nm. An arrow indicates the (maximal)separation as a result of the stearyl layer.

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